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\title{多元统计分析练习4.5-4.7}
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\date{2024 年 5 月 7 日}
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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %Problem 01
设有 $k$ 个总体 $\pi_1,\pi_2,\cdots,\pi_k$, 它们的分布分别是 $N_p(\mu_1,\Sigma), N_p(\mu_2,\Sigma), \cdots, N_p(\mu_k,\Sigma)$. 从这 $k$ 个总体中各自独立地抽取一个样本，取自总体 $\pi_i$ 的样本为 $x_{i1}, x_{i2},\cdots,x_{in_i}$. 考虑假设检验 $$H_0: \mu_1=\mu_2=\cdots=\mu_k, \,\,\, \mathrm{vs.} \,\,\, H_1: \exists i\neq j, \,\mathrm{s.t.}\, \mu_i\neq \mu_j. $$
写出检验统计量和拒绝规则。

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\item  %Problem 02
例子4.5.1. 为了研究销售方式对商品销售额的影响，选择四种商品（甲乙丙丁）按三种不同方式（I，II，III）进行销售。这四种商品的销售额分别为 $x_1,x_2,x_3,x_4$, 数据见表4.5.1. 检验这四种商品的销售额的均值向量，在这三种不同的销售方式下，是否有较大差异。

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\item  %Problem 03
例子4.5.2. 设两个独立样本 $x_1,x_2,\cdots,x_{n_1}$ 和 $y_1,y_2,\cdots,y_{n_2}$ 分别来自多元正态总体 $N_p(\mu_1,\Sigma)$ 和 $N_p(\mu_2,\Sigma)$. 考虑假设检验 $H_0:\mu_1=\mu_2, \,\,\mathrm{vs.}\,\, H_1:\mu_1\neq \mu_2. $ 证明检验统计量 $\Lambda$ 与 $T^2$ 有下述关系式
$$\Lambda = \frac{1}{1+T^2/(n_1+n_2-2)}. $$

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\item  %Problem 04
设有 $k$ 个总体 $\pi_1,\pi_2,\cdots,\pi_k$, 它们的分布分别是 $N_p(\mu_1,\Sigma_1), N_p(\mu_2,\Sigma_2), \cdots, N_p(\mu_k,\Sigma_k)$. 从这 $k$ 个总体中各自独立地抽取一个样本，取自总体 $\pi_i$ 的样本为 $x_{i1}, x_{i2},\cdots,x_{in_i}$. 考虑假设检验 $$H_0: \Sigma_1=\Sigma_2=\cdots=\Sigma_k, \,\,\, \mathrm{vs.} \,\,\, H_1: \exists i\neq j, \,\mathrm{s.t.}\, \Sigma_i\neq \Sigma_j. $$
写出检验统计量和拒绝规则。

\vspace{0.1cm}

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\item  %Problem 05
例子4.6.1. 为了研究销售方式对商品销售额的影响，选择四种商品（甲乙丙丁）按三种不同方式（I，II，III）进行销售。这四种商品的销售额分别为 $x_1,x_2,x_3,x_4$, 数据见表4.5.1. 检验这四种商品的销售额的协方差矩阵，在这三种不同的销售方式下，是否有较大差异。

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\item  %Problem 06
设多元正态总体 $\vec{x}\sim N_p(\mu,\Sigma)$, $\Sigma>0$, 设 $\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n$ 是从总体 $x$ 中抽取的一个简单随机样本。设 $\vec{x}=(x_1,\cdots,x_p)'$. 记 $\rho_{ij}=\rho(x_i,x_j)$. 考虑假设检验
$$H_0: \rho_{ij}=0, \,\,\mathrm{vs.}\,\, H_1: \rho_{ij}\neq 0. $$
写出检验统计量和拒绝规则。


\vspace{0.1cm}

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\item  %Problem 07
设有一个随机变量 $y$ 和一组随机变量 $\vec{x}=(x_1,x_2,\cdots,x_p)'$. 复相关系数 $\rho_{y\cdot\vec{x}}$ 是 $y$ 和线性组合 $\ell'\vec{x}$ 之间的最大相关系数。考虑假设检验
$$H_0: \rho_{y\cdot\vec{x}}=0, \,\,\mathrm{vs.}\,\, H_1: \rho_{y\cdot\vec{x}}\neq 0. $$
写出检验统计量和拒绝规则。


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\item  %Problem 08
设多元正态总体 $\vec{x}=(x_1,\cdots,x_p)'\sim N_p(\mu,\Sigma)$, $\Sigma>0$.  
记 $\vec{x}_1=(x_1,\cdots,x_k)'$, $\vec{x}_2=(x_{k+1},\cdots,x_p)'$.  
记 $\Sigma=\begin{pmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21}& \Sigma_{22} \end{pmatrix}$ 是相应的分块矩阵。
记 $\Sigma_{11\cdot 2} = \Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21} = (\sigma_{ij\cdot k+1,\cdots,p})$
为 $\vec{x}_2$ 是偏变量时，$\vec{x}_1$ 的偏协方差矩阵，这个矩阵的非对角线元素称为偏协方差，对角线元素称为偏方差，偏相关系数定义如下， $$\rho_{ij\cdot k+1,\cdots,p} = \frac{\sigma_{ij\cdot k+1,\cdots,p} }{ \sqrt{\sigma_{ii\cdot k+1,\cdots,p}} \sqrt{\sigma_{jj\cdot k+1,\cdots,p}}}. $$
检验其是否为零，写出检验统计量和拒绝规则。

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\end{enumerate}


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\end{document}

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